Introduction to Ergodic theory Lecture by Amie Wilkinson Notes by Clark Butler October 23, 2014 Let Σ be a finite set. Recall from the previous setup that we have an abelian group G acting on the set ΣG of functions G ! Σ. For a function ! 2 ΣG, this G-action is given by (g · !)(h) = !(g + h); h 2 G For today, G = Z, the integers with the group operation of addition. We will think of Z as parametrizing the action of time. We set Ω := ΣZ. We would like to put an interesting metric on Ω, to study the dynamics on Ω of the shift operator σ :Ω ! Ω, to study σ-invariant measures, and interesting invariant subdynamics of σ. We label the elements of Σ with integers in order to distinguish them. Thus we write Σ = f1; : : : ; ng. Points of Ω will be represented as infinite strings of elements of Σ, together with a decimal point that marks which element of the string corresponds to 0 2 Z. Ω = f! = :::!−2!−1:!0!1!2 ··· : !i 2 Σg Our convention will be that the element to the right of the decimal point corresponds to 0 2 Z. We also use !i to denote the value of the function ! at the integer i. The Z action on Ω has a natural interpretation in this representation. The Z-action is specified by the action of the generator 1. We will denote the action of 1 by σ :Ω ! Ω. σ satisfies the formula [σ(!)](n) = !(n + 1) In the infinite string notation, σ(:::!−2!−1:!0!1!2 ::: ) = :::!−2!−1!0:!1!2 ::: The effect of σ is to shift the decimal point one spot to the right, or equivalently, !0 is shifted one spot to the left. σk shifts the decimal point to the right k spots (for k > 0) and σ−1 shifts the decimal point to the left one spot. A string ! is periodic if there is some k 6= 0 such that σk! = !. All periodic strings of period ` are constructed by taking a finite string j1 : : : j`, ji 2 Σ, placing the decimal point at the beginning of this string, and then concatenating infinitely many copies of this finite string in both directions. Hence it will appear as : : : j1 : : : j`:j1 : : : j`j1 : : : j` ::: We want to put a metric on Ω. Σ carries a natural metric, the discrete metric, for which the distance between i; j 2 Σ is given by δij, the Kronecker delta that is 1 if i = j, and 0 otherwise. Up to scaling, this is the only metric on Σ which is invariant under all permutations of the set f1; : : : ; ng. We define a metric d on Ω by setting, for two strings !; τ, 1 X 1 d(!; τ) = (1 − δ ) 2jij !iτi i=−∞ 1 Exercise: Prove this is a metric on Ω = ΣZ which induces the product topology. The metric d compares the values of τ and ! at each integer, giving more weight to values closer to the decimal point. If τi = !i for jij ≤ m, then 1 d(τ; !) ≤ 2m whereas if τi 6= !i, then 1 d(τ; !) ≥ 2jij So two sequences are close if and only if they agree on all sufficiently small integers. Comment: There is no particular reason for choosing the geometric sequence associated to 1=2 in the definition of d. We could equally well have defined, for any choice of0 < λ < 1, 1 X jij dλ(!; τ) = λ (1 − δ!iτi ) i=−∞ What's important, as we will see later, is that no matter what the choice of λ is, the collection of H¨older continuous functions Ω ! R remains the same (though possibly with different H¨olderexponents depending on λ). A basis for the induced topology of this metric is given by cylinder sets. A cylinder set is a clopen subset of Ω determined by j 2 Z, and a finite word k0; : : : ; k`. C(j; k0; : : : ; k`) = f! 2 Ωj!(j + i) = ki; i = 0; : : : ; `g The integer j centers us at the jth entry of the sequence; C(j; k0; : : : ; k`) is the set of all sequences which start with the word k0 : : : k` from the jth spot. Exercise: Let Σ = f1; 2g have two elements. Let C be the standard middle-thirds Cantor set inside of [0; 1]. Construct a biH¨olderhomeomorphism between Ω and C × C, where C × C carries the induced metric from the L1 metric on R2. Remark: Ω is always a Cantor set, in the sense that there is a homeomorphism from Ω to the standard middle-thirds Cantor set. Any metric space X which is compact, totally disconnected, and perfect is home- omorphic to a Cantor set, and Ω is easily seen to satisfy these properties, no matter what the choice of Σ is. How might one construct an element ! of the shift space with dense orbit, i.e., such that fσk(!): k 2 Zg = Ω? Two procedures were suggested. First, we could enumerate all finite strings of numbers from Σ and concatenate them to form a sequence !. Second, we could take a countable dense collection of points τ j 2 Ω, and form ! by concatenating successively longer strings of numbers from these points τ j. We change perspectives now to look at measures on Ω. Mσ(Ω) = fBorel probability measures µ on Ω such that σ∗µ = µg Recall that σ∗µ = µ is equivalent to saying that µ is invariant under σ, µ(σ−1(A)) = µ(A) = µ(σ(A)) for every Borel set A. The second equality requires that σ is invertible. Invariance in this context means that the measure of a cylinder set C(j; k0; : : : ; k`) does not depend on j. The simplest examples of invariant measures for the shift are given by Bernoulli measures. Let ( n ) n−1 n X ∆ = p 2 R : pi ≥ 0; pi = 1 i=1 2 be the standard n − 1 simplex in Rn. A point p 2 ∆n−1 will be referred to as a probability vector. Given a probability vector p, we define µ on cylinder sets by ` Y µ(C(j; k0; : : : ; k`)) = pki i=0 and setting µ(Ω) = 1, µ(;) = 0. We've defined µ on cylinder sets now; we need to extend µ to be defined on the full Borel σ-algebra of Ω. The tool we will use for this is the Hahn-Kolmogorov extension theorem. Some definitions: For a set X, a σ-algebra is a collection A of subsets of X satisfying 1. ; 2 A 2. A 2 A ! XnA 2 A S 3. For any countable index set I, if Ai 2 A for every i 2 I, then i2I Ai 2 A. S The collection A is called an algebra, if it satisfies 1,2, and the weaker condition 3' that i2I Ai 2 A for any finite index set I. The Hahn-Kolmogorov extension theorem says the following: Let A0 be an algebra, and let A be the smallest σ-algebra containing A0. Let µ0 : A0 ! [0; 1] be a function such that µ0(X) = 1, µ0(;) = 0, and µ0 is finitely additive: for any disjoint sets A1;:::;An 2 A, n ! n [ X µ0 Ai = µ0(Ai) i=1 i=1 Suppose further that µ0 is σ-additive: whenever we have a countable collection fAi 2 Agi2I such that S i2I Ai 2 A, we also have ! [ X µ0 Ai = µ0(Ai) i2I i2I Then µ0 extends to a measure on A. Further, this extension is unique if µ0 is σ-finite. We can check that the measure µ constructed on the algebra generated by the cylinder sets satisfies these properties and so extends uniquely to a measure on the Borel σ-algebra of Ω. The uniqueness property of the Hahn-Kolmogorov extension theorem also means that, to check that µ is σ-invariant, it suffices to check that µ is σ-invariant when restricted to unions of cylinder sets, which is clear. The second example is Markov measures. These are defined by two objects. As before, we have a probability n−1 vector p 2 ∆ , but we also have an n × n stochastic matrix P . P being stochastic means that Pij 2 [0; 1] Pn for every i; j and j=1 Pij = 1. We assume also that p · P = p, so that p is a left eigenvector of P with eigenvalue 1. Written out in sums, this says that n X piPij = pj i=1 for every i; j. P has the property that for every q 2 ∆n−1, q · P 2 ∆n−1. If we take a stochastic matrix P and assume that there is some N > 0 such that all of the entries of P N are strictly positive, then it follows from the Perron-Frobenius theorem that there is a unique left eigenvector p 2 ∆n−1 whose eigenvalue is the leading eigenvalue of P . We can deduce that the leading eigenvalue of P is 1 by observing that the column vector of all 1's is a right eigenvector of P , and then noting that eigenvalues are preserved under transposition. 3 Using the pair (P; p) of a stochastic matrix and an associated probability vector, we can define a measure µ on cylinder sets by `−1 Y µ(C(j; k0; : : : ; k`)) = pk0 · Pkiki+1 i=0 Exercise: Prove that µ gives rise to a shift-invariant measure on Ω. 4.